I just followed a hotly contested discussion at another photographic forum about the simple question of a user whether he should be buying a 35mm or a 50mm lens for his APS-C camera (a D90 btw.).
Soon the term "normal" lens came up and then the discussion centered around what defines a normal lens.
It was quite fun with arguments flying around including the magnification of the viewfinder and I thought to give you the gist and (at last for me) resolve of this discussion.

For an orderly approach we will separate the reasoning around three important aspects, formulated as key-questions here:
1. When is an imaged that a spectator looks at perceived as "normal"?
2. What do you have to do to achieve this impression of normalcy?
3. How does "viewing distance" influence the outcome (2)?

I'm going upstream by suggesting an 18-55 kitlens. It exactly mimics the way we can see: a broad overview of the situation, or a 50 mm view, which is our percieved view when we look at something directly according to most photographers? (that would actually be 18-35 for cropped sensors, pardon me)

_________________I take pictures so quickly, my highschool was "Continuous High".

Now, onto the first question:
When is an imaged that a spectator looks at perceived as "normal"?

Just to make sure that we all understand the same: By "image" I mean a print of a photograph or the display of a photograph on a monitor.
Well here's my definition of a "normal" looking image:
"The image looks like the perfect indistinguishable 2D-rendering of the original 3D-scene."
Or to put it differently and qualifying the word "perfect" in the above definition:
"The objects in the image have the same perceived size as they had in the real world from the point where the image was taken."

You can imagine yourself standing at the point where the photo was taken and someone holds up the perfectly normal image to your eyes at the correct place and distance in front of you and it "blends with the background", so to speak. Your eyes receive the same information whether the image is there or not. It's just like someone made a duplicate of the reality in front of your eyes.

Some remarks:
(1) For the sake of simplicity let's just forget about three-dimensional viewing with both eyes.
(2) The perfect illusion also needs the perfect color reproduction. As this is not influenced by the focal length of the lens you used to capture the image, let's put away with (perfect) colors for the rest of the discussion.
(3) If you take away three-dimensional viewing and color the only important factor for a normal image is the size of the objects you see on the image.
(4) Notice that I spoke of "perceived size" in my second definition. This is very important as I'm not referring to the absolute size of an object in the real world or on the image but the size that this object projects onto our retina (the background of our eyes).
(5) From this, one thing is very clear: If you talk about what you have to do to make a photo normal looking you have to take the viewing distance into account! Because the further away from an image you stand, the smaller all objects in this image look.
(6) This in turn should also caution you about some people saying that there is one focal length that is "normal". You should immediately ask this person about what viewing distances he is assuming. We'll come to this in a later post.

Hehe, you're almost stepping into the "viewfinder-magnification trap", Citruspers!
But it's late for an old man, lemme get some sleep.
Tomorrow we'll have a look at my formula and its components.
But I can already assure all the mathematically not so inclined: You will see only simple multiplications and divisions, no log, no integrals or worse

You know one of the most important lessons I've learned in life is that not all questions are valid. I think this is one of those questions. It's like asking what a square circle looks like...The "normal" lens doesn't exist because nobody can agree upon what "normal" is. If you can define normal a little more clearly then maybe we can talk about the lens that most closely replicates normal.

If the human eye is considered normal, then the lens should have a very small part of the image in focus and be very out of focus around the edges of the image. Most people want sharp focus throughout the entire surface of the lens. The reason why is because in the real world our eyes can move around and focus on different things in the scene. This creates the illusion that we have a lot of things in sharp focus, when really it may only be 10 or 20% of our total vision. The rest is peripheral vision. Our eyes are constantly moving around, surveying the world and creating an image in our brain. It isn't static like a photograph. In other words, it's impossible to have a "normal" lens if what you mean by normal is the human eye. The human eye is connected to our brain which creates a model of the world in our imagination, and until a photograph can change what's in focus based on where we are looking, there will be no normal lens.

That is why I elaborated the point what I see as a valid definition of "normal" in my second post. But perhaps I was not clear enough:
I did not require the image to cover your whole field of view, but only some part: The part (frame) that you wanted to put into your image. But I required to reproduce the perceived size of objects in this frame exactly as you have seen it in the real world.
This concentration on a frame (a "cutout of the real world") is only to make the discussion simpler. As you will see, the formula can also be applied to images that are made of (stitched) panoramas or spheres.

So perhaps you understand why I don't concur with your first paragraph:
- I think the definition is a valid and beautifully simple description of what would be perceived normal
- And the question of how to achieve this "normalcy" is also a valid question

I concede though, that not every photographer in every image tries to achieve normalcy in the sens of the above definition. And it also holds true, that a "normal" picture not necessarily equals a "good" picture.
But that was never the intention in the discussion about "normal" lenses.